12 1 MAP OF THE TERRITORY 1 RESUMO 1 CONTENTS 2 Byzantinisms 2 The meat 5 Parts that were promised and are missing, things that will change 8 BRIEF HISTORY 9 On coauthorship 11 Bibliography A taxonomy of C-systems 15 1 THOU SHALT NOT TRIVIALIZE! Contradictory theories do exist Paraconsistent, but not contradictory! What do you mean? Standing on the shoulders of each other 28 2 A PARACONSISTENT LOGIC IS A PARACONSISTENT LOGIC IS A question of principles The paraconsistency predicament The trivializing predicament Huge tracts of the logical space DEFCON 2: one step short of trivialization C-systems 45 3 COOKING THE C-SYSTEMS ON A LOW FLAME Paleontology of C-systems The basic logic of (in)consistency On what one cannot get Letting bc talk about (dual) inconsistency The logic Ci, where contradiction and inconsistency meet On a simpler presentation for Ci Using LFIs to talk about classical logic 63 viii

13 3.8 Beyond Ci: The dc-systems The opposite of the opposite Consistency may be contagious! Taking it literally: the Brazilian plan completed Algebraic stuff 89 4 FUTUROLOGY OF THE C-SYSTEMS 96 5 REFERENCES 102 Errata to the paper A Taxonomy of C-systems, and more POSSIBLE-TRANSLATIONS SEMANTICS FOR LOGICS OF FORMAL INCONSISTENCY 119 RESUMO DE PTSURVEY 119 RESUMO DE WEAKPTS 119 CONTENTS 120 One size fits all 120 How much is that in real money? 122 Não tem tradução 126 BRIEF HISTORY 128 Bibliography Possible-translations semantics (extended abstract) LOGICS, TRANSLATIONS, POSSIBLE-TRANSLATIONS What is a logic? What is the canonical notion of entailment? What can be done with translations between logics? What are possible-translations semantics? Which logics have adequate semantics? FURTHER ILLUSTRATIONS SOME OTHER RELATED SEMANTIC STRUCTURES 146 REFERENCES 147 ix

14 2.2 Possible-translations semantics for some weak classically-based paraconsistent logics LANGUAGES, BIVALUATIONS, AND SEQUENTS SOME FUNDAMENTAL PARACONSISTENT LOGICS BIVALUED ENTAILMENT, MODALITIES AND MATRICES INTERPRETATIONS THROUGH POSSIBLE TRANSLATIONS ADEQUACY OF EACH OF THE NEWLY PROPOSED PTS 162 REFERENCES MODAL SEMANTICS FOR LOGICS OF FORMAL INCONSISTENCY 169 RESUMO DE LEA 169 RESUMO DE MODPAR 169 RESUMO DE PARANORMAL 170 CONTENTS 171 Some metaphysics Understanding the problem Devising a plan Carrying out the plan Looking back 175 Some esoterism Of algebraization Some puzzles Of replacement Of duality and modality 185 Coda 188 BRIEF HISTORY 189 Bibliography Logics of essence and accident THE WHAT-IT-IS-TO-BE THE MINIMAL LOGIC OF ESSENCE AND ACCIDENT EXTENSIONS OF (K) EA, AND DEFINABILITY OF S AND S 203 x

15 4 CHARACTERIZABILITY OF CLASSES OF FRAMES ON ESSENCE, AND BEYOND 206 ACKNOWLEDGEMENTS 209 REFERENCES Modality and paraconsistency WHAT IS A PARACONSISTENT LOGIC? WHAT IS A MODAL LOGIC? D2 IS NOT A MODAL LOGIC MODAL LOGICS ARE PARACONSISTENT 216 REFERENCES Nearly every normal modal logic is paranormal AFFIRMATIVE AND NEGATIVE MODALITIES Basic modal semantics Modal negations? VARIETIES OF PARANORMALITY Duality, at last The route from modality to paranormality, and the easy way back IMAGINE THERE ARE NO SEA BATTLES 235 ACKNOWLEDGEMENTS 238 REFERENCES AN ABSTRACT PERSPECTIVE ON NEGATION 241 RESUMO DE PURELOCAL 241 RESUMO DE INEFFABLE 241 CONTENTS 242 Here is how you should do it 242 Here is how you should not do it 244 BRIEF HISTORY 248 Bibliography 249 xi

19 Abstract According to the classical consistency presupposition, contradictions have an explosive character: Whenever they are present in a theory, anything goes, and no sensible reasoning can thus take place. A logic is paraconsistent if it disallows such presupposition, and allows instead for some inconsistent yet non-trivial theories to make perfect sense. The Logics of Formal Inconsistency, LFIs, form a particularly expressive class of paraconsistent logics in which the metatheoretical notion of consistency can be internalized at the object-language level. As a consequence, the LFIs are able to recapture consistent reasoning by the addition of appropriate consistency assumptions. So, for instance, while classical rules such as disjunctive syllogism (from A and not-a -or-b, infer B) are bound to fail in a paraconsistent logic (because A and not-a could both be true for some A, independently of B), they can be recovered by an LFI if the set of premises is enlarged by the presumption that we are reasoning in a consistent environment (in this case, by the addition of consistent-a as an extra hypothesis of the rule). The present monograph introduces the LFIs and provides several illustrations of them and of their properties, showing that such logics constitute in fact the majority of interesting paraconsistent systems from the literature. Several ways of performing the recapture of consistent reasoning inside such inconsistent systems are also illustrated. In each case, interpretations in terms of many-valued, possible-translations, or modal semantics are provided, and the problems related to providing algebraic counterparts to such logics are surveyed. A formal abstract approach is proposed to all related definitions and an extended investigation is carried out into the logical principles and the positive and negative properties of negation. Keywords: Universal Logic, negation, paraconsistency, possible-translations semantics, modalities, formal philosophy. xv

27 Prolegomena to Any Future Paraconsistency Some 40 years ago, a remarkable logical approach to the taming of inconsistencies was pioneered by Newton Carneiro Affonso da Costa, in Brazil. The present monograph commemorates this endeavor by updating and extending some chosen aspects of the dacostian approach, centered around the possibility of securing the classical behavior of some assertions made inside a paraconsistent environment. The birth of paraconsistent logic Ai, ai, ai, ai Have you ever danced in the tropics? With that hazy lazy Like, kind of crazy Like South American Way Al Dubin e Jimmy McHugh, South American Way, 1930s. Time often helps us separate the wheat from the chaff. With some luck, inconsequent ideas eventually are abandoned and forgotten. It would be a pity, though, that an important approach to a variety of non-classical logics, and one that is so close to our hearts and minds, would end up remembered only for the wrong reasons. An everlasting myth perpetuated by a considerable parcel of the literature on paraconsistency concerns the alleged origin of paraconsistent logic nearby Curitiba, Paraná, to wit, somewhere in between the pinelands and the sea of Southern Brazil. This section aims at debunking that myth, if only for the sake of intellectual honesty in the practice of the science of logic. Let s initially consider here two expository papers by da Costa and collaborators, namely [43] (1995) and [42] (1999). In [42], for instance, one can find the following assertion [here in my translation]: In fact, the first logician to have built paraconsistent systems having a full scope (propositional logic, predicate logic, set theory) is N. C. A. da Costa (cf. [35], [36]). In a similar vein, [43] mentions, right from the start, the creation of paraconsistent logic by the first author of the present paper [da Costa], more than thirty years ago, as having shown that it is possible to develop a xxiii

28 Prolegomena to Any Future Paraconsistency logic in which contradictions can be mastered, in which there are inoffensive or, at least, not dangerous contradictions. In reality, the paper [43] proposes to tell us the history of the invention of paraconsistent logic, and to that effect it mentions the forerunners of paraconsistent logic (according to the paper: Lukasiewicz, Vasiliev, Jaśkowski, Nelson, Smiley, but not Orlov), setting them at a great distance from the discoverer of paraconsistent logic (according to the paper: Newton da Costa). In the particular case of Vasiliev, the paper asserts that his work was not really understood until the seventies, when the first author [da Costa] read an abstract of a paper of his written in English, and perceived that he had the intuition of paraconsistent logic. Then he suggested that one of his students, A. I. Arruda, investigate Vasiliev s works. On what concerns Jaśkowski, the title of discoverer is denied in that same paper because he has not constructed any discussive logic at the quantificational level. This was done by L. Dubikajtis and the first author [da Costa] in the sixties. Nelson and Smiley are merely mentioned by name, and their works are not commented upon. In addition, a great emphasis seems to be put on the allegation that da Costa developed his paraconsistent calculi in a completely independent way from the works of Vasiliev, Lukasiewicz and Jaśkowski. At that time, in Brazil, the works of these logicians were inaccessible to him. I cannot help but find the above statements utterly puzzling. In his initial thesis on paraconsistent logics, da Costa closes the introduction (p.5) by writing that [in my translation]: Our research had its origin in studies that we have previously published (see [32, 33] and [30, 31]). But, to the best of our knowledge, very little has been done on the topic, besides certain inquiries by Jaśkowski (see [55, 56] and [65, 66]); some studies by Nelson bear some relation to the object of this thesis, though the orientation of the North-American logician is very much distinct from ours (see [68], where you will find bibliographic references). This paragraph alone already seems to seriously impair the last contention about da Costa deserving a special merit for having been a lone researcher with no access to the work of other logicians as he does indeed seem to have had access to all relevant papers, at some point. At any rate, from a historical perspective, why should ignorance or lack of contact with the outside world be attached anything more than a sentimental value at the moment we are assessing one s contribution to science? A more balanced partial account of matters was presented by da Costa himself in his opening address (read by Itala D Ottaviano) at the Stanis law Jaśkowski s Memorial Symposium, held in Toruń, Poland, in 1998 (cf. [40]): I was delighted to notice, in the early 1960 s, that the work I had developed in Brazil by that time had close connections with Jaśkowski s. I recall, as if it were today, reading the English abstract of one of his papers, and realising that the two of us were independently producing xxiv

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